A more realistic scenario is having the direction of gravity towards a center, which is definitely much harder to derive such an equation, and also you will have to redefine the distance traveled as Δθr, assuming that Earth is a perfect sphere with radius(r). However, this only works for the scenario that the direction of gravity is always one direction that is vertically downwards. Hence the equation can be simplified to s = v^2sin(2θ)/g. Lets remind us about the trigonometry identity sin(2θ) = 2cos(θ)sin(θ). Subsititing the equation, getting s = 2v^2sin(θ)cos(θ)/g. From the equation s = vcos(θ)t, and t = 2vsin(θ)/g. Rearranging the equation for finding t, vsin(θ)/g = t, this is the time it takes to reach its maximum height, so we multiply by 2 to get the total time for it to reach the maximum height and return back to the initial height. At maximum height, the vertical velocity(vsin(θ)) is reduced to zero, so the equation should give vsin(θ) - gt = 0. If the ball is moving in a projectile motion, then on reaching the highest point in a flight, the ball will move horizontally and the velocity of the ball in the horizontal motion at this curve path is called the horizontal velocity of the ball. Knowing that the time it takes for the projectile to reach the maximum height from its initial height is the same as the time it takes to fall from the maximum height back to its initial height. In both cases, the ball moves in a horizontal motion. So the issue is to find time(t), the time is affected by the vertical component of velocity and the acceleration due to gravity(g). Knowing that the horizontal velocity = vcos(θ), so we can get the horizontal distance(s) = horizontal velocity x time, s = vcos(θ)t.Ģ. Hence the optimal angle of projection for the greatest horizontal distance is 45° because sin(90) = 1, and any other angle will result in a value smaller than 1.ġ. I tried to drive a formula, ending up having the horizontal distance traveled = v^2sin(2θ)/g. For the question of comparing the horizontal distance traveled of different initial angles of projection.
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